Dynamical Models for the Interaction of Shocks with Dispersive Waves

在这项工作中，获奖者将研究模型的冲击波传播通过分散的环境。这种非线性、多模式的相互作用发生在广泛的自然环境中。一个例子是，水下爆炸产生的声冲击通过海洋传播，并与内部色散波或表面重力波相互作用。另一个例子出现在医疗应用中，其中声波和冲击用于破坏不期望的物体，例如肾结石。第三种情况是超新星爆炸产生的冲击波通过星际等离子体介质传播。尽管这三种物理环境的特点是空间尺度和物理机制截然不同，但它们之间存在着重要的数学相似性。每一个的特点是由一个多分量的振幅方程控制一个快速移动的双曲波（冲击），与一个缓慢移动的色散波非线性相互作用。这种广泛的差异的时间尺度，其特征在于由色散群速度冲击速度的比率，形成一个小参数，它允许一个执行多尺度渐近分析的相互作用过程。 获奖者将关注的物理问题 是当超音速物体在海洋表面上行进时产生的冲击分散波系统的一个例子。 与音爆有关的冲击波在水面上留下压力“足迹”。 感兴趣的是在模拟这种冲击与非线性海洋表面波的相互作用，以了解海洋表面以下的渗透压力。人们怀疑，有限振幅表面波可以起到聚焦“声透镜”的作用，这在原则上可以增加表面以下的噪声水平，并产生更强声能的“热点”。这些问题是海洋学家和海洋生物学家感兴趣的，他们关心海洋表面下传播的噪音量及其对海洋生物的影响。 将被研究的模型方程本身就很有趣，并捕捉了重要的物理机制，需要新的数学工具来分析和理解它们。

In this work, the awardee will study models for the propagation of shock waves through a dispersive environment. These kinds of nonlinear, multi-mode interactions occur in a wide range of natural settings. One example arises when acoustic shocks from an underwater explosion propagate through the ocean and interact with internal dispersive waves or surface gravity waves. Another example arises in medical applications where acoustic waves and shocks are used in the destruction of undesirable objects such as kidney stones. A third arises when shock waves due to supernova explosions propagate through the interstellar plasma media. Despite the fact that these three physical settings are characterized by vastly different spatial scales and disparate physical mechanisms, there are important underlying mathematical similarities that unite them. Each is characterized by a multi-component amplitude equation governing a fast-moving hyperbolic wave (shock), interacting nonlinearly with a slowly-moving dispersive wave. This wide difference of timescales, characterized by the ratio of dispersive group velocity to shock speed, forms a small parameter which allows one to perform a multi-scale asymptotic analysis of the interaction process. The physical problem that the awardee will focus on is an example of a shock-dispersive wave system that arises when a supersonic object travels over the ocean surface. The shock wave associated with a sonic boom leaves a pressure `footprint' on the water's surface. The interest is in modelling the interaction of this shock with the nonlinear ocean surface waves in order to understand the penetration pressure below the ocean surface. It is suspected that the finite amplitude surface waves can act as focusing `acoustic lenses', which in principle can increase the noise level below the surface and create `hot spots' of more intense acoustic energy. Such problems are of interest to oceanographers and marine biologists who are concerned with the amount of noise propagating below the ocean surface and its influence on marine life. The model equations that will be studied are interesting in their own right and capture important physical mechanisms requiring new mathematical tools to analyse and understand them.